Gabor analysis over finite Abelian groups

The topic of this paper are (multi-window) Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time-frequency plane. Our generic approach covers simultaneously multi-dimensional signals as well as non-separable lattices. The main results reduce to well-known fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler-Raz or Tolimieri-Orr, Bastiaans and Van-Leest, among others. In our presentation a central role is given to spreading function of linear operators between finite-dimensional Hilbert spaces. Another relevant tool is a symplectic version of Poisson's summation formula over the finite time-frequency plane. It provides the Fundamental Identity of Gabor Analysis.In addition we highlight projective representations of the time-frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames.Comment: Revised version: two new sections added, many typos fixe

Trace ideals for Fourier integral operators with non-smooth symbols II

We consider Fourier integral operators with symbols in modulation spaces and non-smooth phase functions whose second orders of derivatives belong to certain types of modulation space. We establish continuity and Schatten-von Neumann properties of such operators when acting on modulation spaces.Comment: 25 page

A Deformation Quantization Theory for Non-Commutative Quantum Mechanics

We show that the deformation quantization of non-commutative quantum mechanics previously considered by Dias and Prata can be expressed as a Weyl calculus on a double phase space. We study the properties of the star-product thus defined, and prove a spectral theorem for the star-genvalue equation using an extension of the methods recently initiated by de Gosson and Luef.Comment: Submitted for publicatio

Recent Progress in Shearlet Theory: Systematic Construction of Shearlet Dilation Groups, Characterization of Wavefront Sets, and New Embeddings

The class of generalized shearlet dilation groups has recently been developed to allow the unified treatment of various shearlet groups and associated shearlet transforms that had previously been studied on a case-by-case basis. We consider several aspects of these groups: First, their systematic construction from associative algebras, secondly, their suitability for the characterization of wavefront sets, and finally, the question of constructing embeddings into the symplectic group in a way that intertwines the quasi-regular representation with the metaplectic one. For all questions, it is possible to treat the full class of generalized shearlet groups in a comprehensive and unified way, thus generalizing known results to an infinity of new cases. Our presentation emphasizes the interplay between the algebraic structure underlying the construction of the shearlet dilation groups, the geometric properties of the dual action, and the analytic properties of the associated shearlet transforms.Comment: 28 page

Local well-posedness for the nonlinear Schr\"odinger equation in the intersection of modulation spaces Mp,qs(Rd)∩M∞,1(Rd)M_{p, q}^s(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)

We introduce a Littlewood-Paley characterization of modulation spaces and use it to give an alternative proof of the algebra property, somehow implicitly contained in Sugimoto (2011), of the intersection $M^s_{p,q}(\mathbb{R}^d) \cap M_{\infty, 1}(\mathbb{R}^d)$ for $d \in \mathbb{N}$, $p, q \in [1, \infty]$ and $s \geq 0$. We employ this algebra property to show the local well-posedness of the Cauchy problem for the cubic nonlinear Schr\"odinger equation in the above intersection. This improves Theorem 1.1 by B\'enyi and Okoudjou (2009), where only the case $q = 1$ is considered, and closes a gap in the literature. If $q > 1$ and $s > d \left(1 - \frac{1}{q}\right)$ or if $q = 1$ and $s \geq 0$ then $M^s_{p,q}(\mathbb{R}^d) \hookrightarrow M_{\infty, 1}(\mathbb{R}^d)$ and the above intersection is superfluous. For this case we also reobtain a H\"older-type inequality for modulation spaces.Comment: 14 page

Quantum theta functions and Gabor frames for modulation spaces

Representations of the celebrated Heisenberg commutation relations in quantum mechanics and their exponentiated versions form the starting point for a number of basic constructions, both in mathematics and mathematical physics (geometric quantization, quantum tori, classical and quantum theta functions) and signal analysis (Gabor analysis). In this paper we try to bridge the two communities, represented by the two co--authors: that of noncommutative geometry and that of signal analysis. After providing a brief comparative dictionary of the two languages, we will show e.g. that the Janssen representation of Gabor frames with generalized Gaussians as Gabor atoms yields in a natural way quantum theta functions, and that the Rieffel scalar product and associativity relations underlie both the functional equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change

The finiteness of the four dimensional antisymmetric tensor field model in a curved background

A renormalizable rigid supersymmetry for the four dimensional antisymmetric tensor field model in a curved space-time background is constructed. A closed algebra between the BRS and the supersymmetry operators is only realizable if the vector parameter of the supersymmetry is a covariantly constant vector field. This also guarantees that the corresponding transformations lead to a genuine symmetry of the model. The proof of the ultraviolet finiteness to all orders of perturbation theory is performed in a pure algebraic manner by using the rigid supersymmetry.Comment: 23 page

Endogenous Growth of Population and Income Depending on Resource and Knowledge

We consider a three sector demoeconomic model and its interdependence with the accumulation of human capital and resources. The primary sector harvests a renewable resource (fish, corn or wood) which constitutes the input into industrial production, the secondary sector of our economy. Both sectors are always affected by the stock of knowledge. The tertiary sector (schooling, teaching, training, research) is responsible for the accumulation of this stock that represents a public good for all three sectors. Labor is divided up between the three sectors under the assumption of competitive labor markets. A crucial feature of this economy is the importance of public goods--stock of knowledge and the common--which requires collective actions. Absence of collective actions describes the limiting case of hunters and gatherers. The central focus of this study is whether and what kind of interactions between the economy, the population and the environment foster sustainability and, if possible, continuous growth

Fertilization and early embryology: Use of lasers in assisted fertilization and hatching

The erbium-yttrium-aluminium-garnet (Er: YAG) laser has been applied to micromanipulation in humans. It was used in the fertilization process for both subzonal insemination (SUZI) and for partial zona dissection (PZD). Laser-assisted micromanipulation achieved significantly higher fertilization rates (34.8%) when compared to mechanical SUZI (16.1%), but use of the laser did not improve the PZD results (laser 14.8% versus mechanical 14%). The Er: YAG laser was used to assist hatching. In the mouse it significantly improved the hatching rate (80 versus 29.3%) 110 h after administration of human chorionic gonadotrophin. This technique was applied in two different centres to patients with previous in-vitro fertilization (IVF) failures. The implantation rate per embryo (14.4% laser-assisted hatching versus 6% control group) and the pregnancy rate per transfer (40 versus 16.2%) were improve

A Guide to Localized Frames and Applications to Galerkin-like Representations of Operators

This chapter offers a detailed survey on intrinsically localized frames and the corresponding matrix representation of operators. We re-investigate the properties of localized frames and the associated Banach spaces in full detail. We investigate the representation of operators using localized frames in a Galerkin-type scheme. We show how the boundedness and the invertibility of matrices and operators are linked and give some sufficient and necessary conditions for the boundedness of operators between the associated Banach spaces.Comment: 32 page